Application of Multilevel Models in Dentistry.

Multilevel analysis which was primarily introduced to deal with hierarchical data was later applied extensively for research in other fields of science and not only for nested data, but also for repeated measurements or clustered trials. This method of statistical analysis was applied in dental studies in the 1991 for the first time but despite its value for data analysis in dental studies, its application for dental studies remains limited until now. This manuscript reviews the applications of this method in dental studies.


INTRODUCTION
Multilevel analyses are attempted to find correlations among variables that have a hierarchical or nested nature. They are aimed at eliminating the problems due to the shortcomings of traditional analyses in perceiving the correlations among variables at different levels [1]. This type of analysis was first introduced by Aitkin et al, [2] in the early 1980 for the purpose of educational assessment for nested data. It was later applied extensively for research in other fields of science and not only for nested data, but also for repeated measurements or clustered trials [3]. In dentistry, multilevel models were first applied in 1991 for the assessment of craniofacial growth curves for orthodontic purposes [4]. Although the nature of most data in dental research is compatible with such analyses, they have been rarely used in studies. Figure 1 shows the number of dental articles published in PubMed during 1991-2013 that used multilevel analysis. In the mentioned time period, multilevel analysis was used in only 233 articles; out of which, 42 had been published in non-dental and 191 in 45 dental journals. Number of dental and non-dental journals that published multilevel model dental papers in this time period is shown in Figure 2. A noteworthy issue is that of 191 papers published in dental journals, 155 had been published in specialty journals in the fields of periodontics, orthodontics and oral health and community dentistry and a total of 184 articles were in the mentioned three fields. Also, 91 articles had been published in six and 123 papers in 10 specific journals. Therefore, it seems that using multilevel analysis is by invitation and demand of reviewers and editors of specific journals familiar with this method rather than being the choice of researchers. All that said, with the presumption of lack of familiarity of dental researchers with multilevel analysis and its applications despite the passage of many years since its introduction, this study sought to reintroduce this model by setting examples of its different applications in dental research. Although the aim of this study is not providing the mathematical model of multilevel analysis or discussing its different types and characteristics, for the purpose of primary acquaintance, we offer a simplified definition of this model. In unilevel (ordinary) regression model, the following formula is used to assess the effect of variable X on factor Y. Yi = β0 + β1Xi + ei Where β0 is a fixed amount, β1 is the slope in the conditional regression model for Y |X and e is the estimated error rate. If in this model, several Xj (from X1 to Xn) are used instead of one independent variable, the model will be in the following form: Yi = β0 + ∑βjXij + ei Assuming that in studies with hierarchical data, a higher-level variable affects the lower-level ones, the equation can be written in such way that all elements in the equation are influenced by the upper level variable. It means that, for each amount of the higher level variable, the coefficients of the equation will change. In other words, instead of one equation, we will have a separate equation for each amount of the higher-level variable. Assume that in your study, you want to assess the effect of oral hygiene instruction (ME) on plaque index (PI) of students. To simplify the model in this study, we only consider one covariate i.e. age. In this case, the suggested model for unilevel regression analysis would be as follows: Yi = β0 + β1age + β2ME + ei If this study was carried out as clustered in different schools, in order to assess the effect of type of school in the model, for each of the β0, β1 and β2 coefficients in the multilevel equation, the effect of type of school variable (S) would also be taken into account as follows: β0 = δ00 + δ01Sj + U0j β1 = δ10 + δ11Sj + U1j β2 = δ20 + δ21Sj + U2j Thus, for each type of school, a different equation is created to assess the effect of type of intervention on PI of students at different ages. Such change in coefficients of the equation can well justify the difference in the efficacy of education in different schools. One advantage of this model is that by using a specific type of this model known as variance component model, we can find out the percentage of the changes of a dependent variable that occur at each level. In the aforementioned example, it can be found that how much of the change in PI is due to the individual changes and how much is due to differences between students of different schools [3].

Hierarchical data:
In dentistry, similar to other fields of science, health-related outcomes and health or disease status occur as the result of the interaction effect between the individuals and their surrounding environment. These factors are known as the social determinants of health and in order to recognize them and evaluate their effect on oral health, multilevel models must be applied.
In the domains of oral health and community dentistry, social factors are important determinants of the risk of tooth decay. "There are several reasons why neighborhood environment may affect dental caries in negative and positive ways. Availability of, and access to, healthy foods may differ across areas, as may availability and access to dental care" [5].
Detecting the effect of these factors and their interaction with individual risk factors requires multilevel analysis. Tellez et al, [5] in their study found that the likelihood of caries development was lower in areas with higher number of churches; whereas, risk of caries development increased in areas with higher number of grocery stores. However, in some investigations, despite the conduction of multilevel analysis, the effect of factors in higher than individual levels is found to be insignificant and the analysis eventually turns into a unilevel regression analysis [6]. Nonetheless, first the effect of social factors must be assessed in multilevel models because the magnitude of the effects of factors at the social level is variable in different communities and must be measured. Moreover, the effect of variables at both individual and social levels on developing hygiene habits like tooth brushing can be evaluated in such studies. For instance, a study conducted in Pretoria in 2009 demonstrated that the effect of school grade variable on frequency of tooth brushing decreased while the age increased from 11 to 15 [7]. Multilevel modeling has also been applied to estimate the effect of individual and social parameters on the likelihood of developing periodontal disease [8] or the prevalence of pain [9,10]. These models are especially important when assessing the effect of social factors on diseases, in which environmental factors play a more serious role like fluorosis [11]. Both individual and social factors play a role in development of caries but despite the fact that all teeth are located in the same environment, the oral cavity, carious lesions occur in some of them and other teeth remain intact, at least by the time of examination. Such differences confirm the theory of the effect of some factors at a lower than individual level on development of caries. These variables can be related to the anatomical form of the teeth, their location in the jaw and dental arch, class of malocclusion and even presence of crowding; because the location of accumulated microbial plaque also affects the efficacy of plaque removal. Presence of such variables necessitates the use of multilevel models to study the behavior of tooth decay and investigate the effect of variables at lower than individual levels namely at the level of tooth or even tooth surface [12]. Barga  In some longitudinal studies, data at each time point have a hierarchical nature. In such cases, time, as the lowest level, can be entered into the multilevel regression model [31]. Gilthorpe et al.
[32] conducted a longitudinal study on periodontal patients to test the accuracy of two theories regarding the mechanism of periodontal disease progression. They applied a 4-level (repeated measurement, site, tooth, subject) multilevel model to assess the trend of changes in pocket depth. The understudy sample can be human, animal or even laboratory experiments, and multilevel models can be used as long as several follow ups are made. A few recent studies have used multilevel models for the analysis of data with several follow up sessions in animal models. For instance, Liu et al, [33] in 2010 used this method of analysis to compare the effect of continuous and intermittent load on suture expansion in New Zealand white juvenile male rabbits at different time points. This method has also been used in in vitro studies to compare the microbial colony count in the water flow of dental units at different time points [34]. In laboratory study designs or repeated measures, crossover or paired-data trials, multilevel models can be applied based on the assumption of repeated data of each specimen at two time points or two different areas of a specimen. Tang et al.
[35] used this method of analysis to compare the masticatory function in two-and four-implant supported overdentures. In a crossover, randomized trial, they fabricated two types of overdentures for patients and compared the masticatory function of patients in the two groups at three different time points with five foods. In this three-level model, the tested food products comprised level 1, time of assessment comprised the 2 nd level and the overdenture design was the 3 rd level. A similar approach was used in a split-mouth animal study [36]. In many cases, both multilevel and unilevel repeated measure models can be used for the analysis of repeated data. Even in many cases, unilevel repeated measure analyses are preferred to multilevel models due to their easy application and interpretation. However, the weakness of unilevel model is the necessity of presence of data at all follow ups. If a sample does not show up in one follow up, that specimen must be totally excluded from the analysis or that specific data must be imputed with specific methods.

Diagnostic studies and validity measurements:
In studies assessing clinical diagnostic techniques, particularly for caries detection, data have a hierarchical nature; because when comparing diagnostic methods, whether the aim is to assess validity or reliability, assessments are done on single series of samples; which makes the comparison of different techniques difficult with the conventional analytical methods. In caries detection clinical studies, the reproducibility of caries detection can be evaluated at the three levels of individual, tooth and surface [39].

Systematic reviews:
In systematic reviews, researchers try to combine the results of different studies in order to draw a general conclusion. Thus, considering the two levels of sample and study, multilevel models can be applied for data analysis. This method has one main advantage over the conventional metaanalysis methods; that is, studies reporting different results do not have to be necessarily excluded in case of heterogeneity of results; because all studies can be presented in a bi-level model [40].
In a systematic review aiming to assess orthodontic bond strength in-vitro, since the primary studies had been conducted by a few specific authors, the three-level model was used for data analysis [41].

CONCLUSION
Introduction of multilevel models, due to their extensive applications, has revolutionized the analysis of study results in the fields of social sciences as well as medical sciences. In medicine, these models can be applied to all types of data retrieved from a wide range of studies from epidemiologic and descriptive to clinical trails and even laboratory experimental studies. By the advances in different multilevel models, they can now be applied to different outcome variables with linear regression, logistic regression [31], Poisson [42], negative binomial [43] and even survival data [23] models. At first, limited software programs were introduced for these analyses; but, at present, the majority of multilevel analyses can be performed by some software programs like SAS and R. Simpler multilevel models can even be accessed in conventional statistical software programs like SPSS and STATA. The noteworthy issue here, is the introduction of these methods, their principles, applications and advantages for data analysis and particularly presenting models that better fit the reality of biomedical phenomena than single level models.

ACKNOWLEDGMENTS
This study was part of a PhD thesis supported by Tehran University of Medical Sciences.